In previous studies, spectral winding numbers have been introduced
to depict the transitions between different NHQC
phases [1]}. The general idea is to incorporate a periodic
parameter \(\theta \)  into the Hamiltonian \(H\)  of the system, usually
achieved by imposing the twist boundary condition (flux insertion),
and then track the spectral flow of the parameter-dependent Hamiltonian
\(H(\theta )\)  with respect to a certain base energy on the complex
plane during the change of \(\theta \)  over a cycle. If the spectrum
of \(H(\theta )\)  is real, all eigenvalues collapse onto the real
axis and their winding number \(w\)  must vanish with respect to any
base energies. If \(H(\theta )\)  possesses complex energies, their
flow with respect to \(\theta \)  could form loops around certain points
on the complex plane, and the spectral winding number of \(H(\theta )\) 
around these points could be nonzero and quantized. Therefore, the
winding number \(w\)  can be employed as a topological order parameter
to distinguish NHQC phases with real and complex eigenspectrum.
